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Key Highlights
- Frequency polygons are graphical representations of statistical data that help understand complex data easily.
- It is considered identical to the histogram.
- Frequency polygons are used to compare data sets or illustrate a cumulative frequency distribution.
- The graph can be used to represent discrete and continuous series of data.
- It uses a linear graph to represent quantitative data and its frequencies.
In statistics, frequency polygons involve the representation of data graphically, where frequencies of classes are plotted against their midpoints.
- It is roughly identical to a histogram, which is used to compare data sets or display a cumulative frequency distribution.
- Frequency polygon and a histogram can be used together to obtain a more accurate view of the distribution shape.
- When plotting a frequency polygon graph, calculate the midpoint or the classmark for every class interval.
- The formula for the same is:
Class Mark (Midpoint) = (Upper Limit + Lower Limit) / 2
Key Terms: Frequency Polygons, Histogram, Statistics, Frequency Polygon Formula, Cumulative Frequency Distribution, Class Interval
Frequency Polygons
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Frequency polygons can be defined as a graphical representation of data distribution which helps in understanding the data via the help of a specific shape.
- The graph helps to show cumulative frequency distribution data by means of a line graph.
- Frequency polygons are very similar to histograms and help when comparing two or more data.
- They are expressed in the form of graphs which interprets information used in statistics.
- The visual data help depict the trend and shape of data in an organised and systematic manner.
- Frequency Polygons show the number of occurrences of class intervals via the shape of the graph.
- While a frequency graph is a line graph showing cumulative frequency distribution data, a histogram shows a graph with rectangular bars without spaces.
Types of Frequency Polygon
There are several types of frequency polygons, which include:
| Type | Description |
|---|---|
| Absolute Frequency Polygon | Absolute frequency polygon has peaks showing the actual number of points in the associated interval. |
| Relative Frequency | It is a histogram graph showing the relative frequency or probability density of a single variable. |
| Relative Frequency Polygon | A relative frequency polygon has peaks showing the percentage of total data points that fall within the interval. |
Important Terms of Frequency PolygonsSome terms of Frequency Polygons are:
|
Frequency Polygon
Steps to Draw Frequency Polygon
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The following are the steps required to draw frequency polygons:
- Select a class interval and record the results on the horizontal axes.
- Mark the midpoint of every interval at the horizontal axes.
- On vertical axes, mark the frequency.
- Mark a point at the height of each class interval at the centre of each class interval that corresponds to the frequency of the class interval.
- Join these lines to form a line segment.
- The figure obtained after joining the lines is known as a frequency polygon.
Precaution While Drawing The Curves:
- The points should be shown in a way that they are straight.
- It is necessary to draw rectangles because it shows a wider image of the data that is going to be represented on the graph.
Frequency Polygons Formula
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When a frequency polygons graph are plotted, the midpoint or the classmark for each of the class intervals is calculated. Thus, the formula for calculating Classmark for each interval is:
Classmark = (Upper limit + Lower limit) / 2
Frequency Polygons ExampleAn example of a frequency polygons can be shown as: Create a frequency polygon using the following information:
Solution: First we need to calculate the cumulative frequency of the given data-
First, we have to begin plotting the class, such as 54.5, 64.5, 74.5, and so on, all the way up to 94.5. Here, the previous and next class marks to begin and end the polygon, i.e. 44.5 and 104.5. The frequencies related to each classmark are then plotted against each class mark. Thus, the frequency for class marks 44.5 and 104.5 is zero and touches the x-axis, as shown below. These plot points are simply utilized to give the polygon a closed shape. This is how the polygon looks:  Test Scores Example |
Advantages of Frequency Polygons
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The advantages of frequency polygons are as follows:
- The frequency polygon not only sorts and represents the data but also makes it easy to compare the given data.
- They are easy to understand and provide a clear view of the distribution data.
- This method is also less time-consuming as compared to other methods.
- It's great for comparing two pieces of data that are of the same type, especially when the data is vast and continuous.
Difference Between Frequency Polygons and Histogram
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The difference between frequency polygons and histograms are:
| Frequency Polygon | Histograms |
|---|---|
| A Frequency polygon graph is shown in the form of a curve denoted by a line segment. | A histogram is a graph which shows data via rectangular-shaped bars with no space between them. |
| Here, the midpoint of the frequencies is used. | The frequencies are seen to be spread evenly over the class intervals. |
| The accurate points in a frequency polygon graph demonstrate data of the given class interval. | The data comparison is not visually appealing in a histogram graph. |
Sample Questions
Ques. What is the difference between a frequency curve and a frequency polygon? (2 marks)
Ans. The difference between a frequency polygon and a frequency curve is that a frequency polygon is drawn by connecting points with a straight line, but a frequency curve is drawn with a free hand. A frequency polygon is very closely similar to a histogram. While, a frequency curve can be obtained by joining the top midpoints of all rectangles of a histogram by means of free hand.
Ques. What is the formula of the Frequency Polygon Graph? (1 mark)
Ans. The Formula of the Frequency Polygon Graph is: Class Mark (Midpoint) = (Upper Limit + Lower Limit) / 2.
Ques. Why is a frequency polygon used? (2 marks)
Ans. Frequency polygons help us grasp the shape of the data as well as its tendencies. They act like histograms but are helpful when comparing sets of data. Frequency polygons help display cumulative frequency distributions.
Ques. What is Relative Frequency Polygon? (2 marks)
Ans. A Relative Frequency Polygon can be expressed as a graph with peaks showing the percentage of total data points that fall within the interval. For example, in a test score example, the relative frequency can be calculated by dividing the frequency of a score by the number of scores (N).
Ques. What are the differences between Frequency Polygon and Histogram? (2 marks)
Ans. The differences between Frequency Polygon and Histogram are:
| Frequency Polygon | Histograms |
|---|---|
| A Frequency polygon graph is a form of a curve denoted by a line segment. | A histogram is a graph showing data via rectangular-shaped bars with no space between them. |
| The midpoint, here, of the frequencies is used. | Here, the frequencies spread evenly over the class intervals. |
Ques. What are the uses of frequency polygons? (2 marks)
Ans. Some uses of Frequency Polygons are:
- Frequency polygons are used when the given data is large, and continuous in form.
- It is used while comparing two different sets of data of the same nature.
Ques. What are the steps involve in constructing a frequency polygon? (3 marks)
Ans. For making the polygraph following steps are needed to be followed-
- Make a histogram
- Calculate the midpoint of each bar.
- Placing a point at the beginning and end of the histogram
- Connect the points.
Ques. A frequency distribution's frequency polygon is depicted below.
Answer the following question based on this:
1. What is the average number of students in a class period with a class mark of 15?
2. What is the class interval for a 45th grader?
3. For the distribution, construct a frequency table. (3 marks)
Ans. The following answers can be shown as:
- 18
- 40-50
- The class intervals are 0 – 10, 10 – 20, 20 – 30, 30 – 40, 40 – 50, 50 – 60 since the class marks of consecutive overlapping class intervals are 5, 15, 25, 35, 45, 55. As a result, the frequency table is built as follows.
| Class Interval | Frequency |
|---|---|
| 0 - 10 | 10 |
| 10 - 20 | 18 |
| 20 - 30 | 14 |
| 30 - 40 | 26 |
| 40 - 50 | 8 |
| 50 - 60 | 18 |
Ques. If the weight range for a class of 45 students is distributed by 30 - 40, 40 - 50, 50 - 60, 60 - 70. What would be the class marks for each weight range? (3 marks)
Ans. Calculating the classmark for a frequency polygon graph, we use the formula, Classmark = (Upper Limit + Lower Limit) / 2.
Hence,
Class interval 30 - 40 = (40 + 30)/2 = 35
Class interval 40 - 50 = (50 + 40)/2 = 45
Class interval 50 - 60 = (60 + 50)/2 = 55
Class interval 60 - 70 = (70 + 60)/2 = 65
Ques. What are the features of frequency polygons? (2 marks)
Ans. The features of frequency polygons are as follows:
- Frequency polygons emphasize the central tendency of the data by connecting the midpoints of the intervals.
- Compared to histograms with rectangular bars, frequency polygons provide a smoother visual representation of the data's distribution.
- This can be particularly useful for comparing data sets with similar ranges but potentially different distributions within those ranges.
Ques. If the weight range for a class of 60 students is distributed by 70 - 80, 80 - 90, 90 - 100, 100 - 110. What would be the class marks for each weight range? (3 marks)
Ans. Calculating the classmark for a frequency polygon graph, we use the formula, Classmark = (Upper Limit + Lower Limit) / 2.
Hence,
Class interval 70 - 80 = (70 + 80)/2 = 75
Class interval 80 - 90 = (80 + 90)/2 = 85
Class interval 90 - 100 = (90 + 100)/2 = 95
Class interval 100 - 110 = (100 + 110)/2 = 105
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